MinT - Best Known (217−81, 217, s)-Nets in Base 3

Best Known (217−81, 217, s)-Nets in Base 3

(217−81, 217, 156)-Net over F₃ — Constructive and digital

Digital (136, 217, 156)-net over F₃, using

11 times m-reduction [i] based on digital (136, 228, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 114, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(217−81, 217, 245)-Net over F₃ — Digital

Digital (136, 217, 245)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(217−81, 217, 2933)-Net in Base 3 — Upper bound on s

There is no (136, 217, 2934)-net in base 3, because

1 times m-reduction [i] would yield (136, 216, 2934)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 455174 666788 718625 765439 265971 159230 044865 985092 451128 172947 697889 625131 105410 603649 142822 932654 366737 > 3²¹⁶ [i]